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Numerical Solution of Time and Space Fractional Burger's Equation with Finite Difference Method

Year 2019, , 71 - 83, 31.08.2019
https://doi.org/10.30931/jetas.614506

Abstract

In this study, fractional Burger’s Equation, which has Dirichlet Boundary Conditions, is solved with the Finite Difference Method. Fractional Burger Equation is found by S. Momani, which is made with changing time and space terms with fractional terms. This equation is solved with the finite difference method and analysis of this scheme is discussed with examples. Stability and Uniqueness are discussed with using matrix method. We compare analytical and numerical solutions with error analysis of them.

References

  • [1] Zhang, Y., “A finite difference method for fractional partial differential equation”, Applied Mathematics and Computation 215 (2009) : 524-529.
  • [2] Podlubny, I., “Fractional differential equation”, V198 of Mathematics in Science and Engineering, Academic Press, San Diego, (1999).
  • [3] Zhang, P.G., Wang, J.P., “A predictor–corrector compact finite difference scheme for Burgers’ equation”, Applied Mathematics and Computation 219 (2012) : 892-898.
  • [4] Varöglu, E., Liam Finn, W.D., “Space-time finite elements incorporating characteristics for the burgers’ equation”, Int. J. Numer. Methods Eng. 16 (1980) : 171–184.
  • [5] Kutluay, S., Bahadir, A.R., Özdes, A., “Numerical solution of one-dimensional burgers equation: explicit and exact-explicit finite difference methods”, J. Comput. Appl. Math 103 (1999) : 251–261.
  • [6] Pandey, K., Verma, L.,. Verma, A.K, “On a finite difference scheme for burgers’ equation”, Appl. Math. Comput 215 (2009) : 2206-2214.
  • [7] Liao, W., “An implicit fourth-order compact finite difference scheme for one-dimensional burgers’ equation”, Appl. Math. Comput 206 (2008) : 755-764.
  • [8] Saria, M., Gürarslanb, G., “A sixth-order compact finite difference scheme to the numerical solutions of burgers’ equation”, Appl. Math. Comput 208 (2009) : 475-483.
  • [9] Hon, Y., Mao, X., “An efficient numerical scheme for burgers’ equation”, Appl. Math. Comput 95 (1998) : 37-50.
  • [10] Asaithambi, A., “Numerical solution of the burgers’ equation by automatic differentiation”, Appl. Math. Comput 216 (2010) : 2700-2708.
  • [11] Mena, S.A., Salim, B.C., “Ettahlil-lül adedil muadeleti burger biistihdamil frukatil müntahiyeti”, Mecelle-tül rafdin liulumil muhasebeti verriyaziyatil mücellidi, (2007) : 1-4.
  • [12] Burgers, J.M., “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech 1 (1948) : 171-199.
  • [13] Kurulay, M., Bayram, M., “Some properties of the Mittag-Leffler functions and their relation with the Wright functions”, Advances in Difference Equations (2012) : 181.
  • [14] Kurulay, M., “The approximate and exact solutions of the space- and time-fractional Burgers equations”, International Journal of Research and Reviews in Applied Sciences, 3 (2010) : 257-263.
  • [15] Momani, S., “Non-perturbative analytical solutions of the space and time fractional Burgers equations”, Chaos, Solutions and Fractals, 28 (4) (2006) : 930-937.
  • [16] Bashan, A., Karakoc, S.B.G., Geyikli, T., “Approximation of the KdVB equation by the quintic B-spline differential quadrature method”, Kuwait journal of science, 42 (2) (2015) : 67-92.
  • [17] Yılmaz, S., Ünlütürk Y., Mağden, A., “A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space”, SAÜ Fen Bil Der 20 (2) (2016) : 325-335.
  • [18] Meerschaert, M.M., Tadjeran, C., “Finite difference approximations for fractional advection–dispersion flow equations”, J. Comput. Appl. Math. 172 (2003) : 65-77.
  • [19] Meerschaert, M.M., Scheffler, H.P., Tadjeran, C., “Finite difference method for two dimensional fractional dispersion equation”, J. Comput. Phys. 211 (2006) : 249–261.
Year 2019, , 71 - 83, 31.08.2019
https://doi.org/10.30931/jetas.614506

Abstract

References

  • [1] Zhang, Y., “A finite difference method for fractional partial differential equation”, Applied Mathematics and Computation 215 (2009) : 524-529.
  • [2] Podlubny, I., “Fractional differential equation”, V198 of Mathematics in Science and Engineering, Academic Press, San Diego, (1999).
  • [3] Zhang, P.G., Wang, J.P., “A predictor–corrector compact finite difference scheme for Burgers’ equation”, Applied Mathematics and Computation 219 (2012) : 892-898.
  • [4] Varöglu, E., Liam Finn, W.D., “Space-time finite elements incorporating characteristics for the burgers’ equation”, Int. J. Numer. Methods Eng. 16 (1980) : 171–184.
  • [5] Kutluay, S., Bahadir, A.R., Özdes, A., “Numerical solution of one-dimensional burgers equation: explicit and exact-explicit finite difference methods”, J. Comput. Appl. Math 103 (1999) : 251–261.
  • [6] Pandey, K., Verma, L.,. Verma, A.K, “On a finite difference scheme for burgers’ equation”, Appl. Math. Comput 215 (2009) : 2206-2214.
  • [7] Liao, W., “An implicit fourth-order compact finite difference scheme for one-dimensional burgers’ equation”, Appl. Math. Comput 206 (2008) : 755-764.
  • [8] Saria, M., Gürarslanb, G., “A sixth-order compact finite difference scheme to the numerical solutions of burgers’ equation”, Appl. Math. Comput 208 (2009) : 475-483.
  • [9] Hon, Y., Mao, X., “An efficient numerical scheme for burgers’ equation”, Appl. Math. Comput 95 (1998) : 37-50.
  • [10] Asaithambi, A., “Numerical solution of the burgers’ equation by automatic differentiation”, Appl. Math. Comput 216 (2010) : 2700-2708.
  • [11] Mena, S.A., Salim, B.C., “Ettahlil-lül adedil muadeleti burger biistihdamil frukatil müntahiyeti”, Mecelle-tül rafdin liulumil muhasebeti verriyaziyatil mücellidi, (2007) : 1-4.
  • [12] Burgers, J.M., “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech 1 (1948) : 171-199.
  • [13] Kurulay, M., Bayram, M., “Some properties of the Mittag-Leffler functions and their relation with the Wright functions”, Advances in Difference Equations (2012) : 181.
  • [14] Kurulay, M., “The approximate and exact solutions of the space- and time-fractional Burgers equations”, International Journal of Research and Reviews in Applied Sciences, 3 (2010) : 257-263.
  • [15] Momani, S., “Non-perturbative analytical solutions of the space and time fractional Burgers equations”, Chaos, Solutions and Fractals, 28 (4) (2006) : 930-937.
  • [16] Bashan, A., Karakoc, S.B.G., Geyikli, T., “Approximation of the KdVB equation by the quintic B-spline differential quadrature method”, Kuwait journal of science, 42 (2) (2015) : 67-92.
  • [17] Yılmaz, S., Ünlütürk Y., Mağden, A., “A study on the characterizations of non-null curves according to the Bishop frame of type-2 in Minkowski 3-space”, SAÜ Fen Bil Der 20 (2) (2016) : 325-335.
  • [18] Meerschaert, M.M., Tadjeran, C., “Finite difference approximations for fractional advection–dispersion flow equations”, J. Comput. Appl. Math. 172 (2003) : 65-77.
  • [19] Meerschaert, M.M., Scheffler, H.P., Tadjeran, C., “Finite difference method for two dimensional fractional dispersion equation”, J. Comput. Phys. 211 (2006) : 249–261.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Muhammet Kurulay 0000-0002-9276-9989

İbrahim Şentürk 0000-0003-2914-9119

Publication Date August 31, 2019
Published in Issue Year 2019

Cite

APA Kurulay, M., & Şentürk, İ. (2019). Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method. Journal of Engineering Technology and Applied Sciences, 4(2), 71-83. https://doi.org/10.30931/jetas.614506
AMA Kurulay M, Şentürk İ. Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method. JETAS. August 2019;4(2):71-83. doi:10.30931/jetas.614506
Chicago Kurulay, Muhammet, and İbrahim Şentürk. “Numerical Solution of Time and Space Fractional Burger’s Equation With Finite Difference Method”. Journal of Engineering Technology and Applied Sciences 4, no. 2 (August 2019): 71-83. https://doi.org/10.30931/jetas.614506.
EndNote Kurulay M, Şentürk İ (August 1, 2019) Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method. Journal of Engineering Technology and Applied Sciences 4 2 71–83.
IEEE M. Kurulay and İ. Şentürk, “Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method”, JETAS, vol. 4, no. 2, pp. 71–83, 2019, doi: 10.30931/jetas.614506.
ISNAD Kurulay, Muhammet - Şentürk, İbrahim. “Numerical Solution of Time and Space Fractional Burger’s Equation With Finite Difference Method”. Journal of Engineering Technology and Applied Sciences 4/2 (August 2019), 71-83. https://doi.org/10.30931/jetas.614506.
JAMA Kurulay M, Şentürk İ. Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method. JETAS. 2019;4:71–83.
MLA Kurulay, Muhammet and İbrahim Şentürk. “Numerical Solution of Time and Space Fractional Burger’s Equation With Finite Difference Method”. Journal of Engineering Technology and Applied Sciences, vol. 4, no. 2, 2019, pp. 71-83, doi:10.30931/jetas.614506.
Vancouver Kurulay M, Şentürk İ. Numerical Solution of Time and Space Fractional Burger’s Equation with Finite Difference Method. JETAS. 2019;4(2):71-83.